Decompositions of Ehrhart $h^*$-polynomials for rational polytopes

Abstract

The Ehrhart quasipolynomial of a rational polytope $P$ encodes the number of integer lattice points in dilates of $P$, and the $h^*$-polynomial of $P$ is the numerator of the accompanying generating function. We provide two decomposition formulas for the $h^*$-polynomial of a rational polytope. The first decomposition generalizes a theorem of Betke and McMullen for lattice polytopes. We use our rational Betke--McMullen formula to provide a novel proof of Stanley's Monotonicity Theorem for the $h^*$-polynomial of a rational polytope. The second decomposition generalizes a result of Stapledon, which we use to provide rational extensions of the Stanley and Hibi inequalities satisfied by the coefficients of the $h^*$-polynomial for lattice polytopes. Lastly, we apply our results to rational polytopes containing the origin whose duals are lattice polytopes.

Figures (2)

• Figure 1: This figure shows $\mathrm{cone}\left(P\right)$ (in orange), $P$, $3P$, $(\mathbf{a},\ell)=(2,4)$, $\mathrm{Box}\left(\Delta_1'\right)$ (in yellow), $\mathrm{Box}\left(\Delta_2'\right)$ (in pink).
• Figure 2: The rational hexagon $P^*_\mathcal{L}$.

Theorems & Definitions (29)

• Example 2.1
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3: Stanley Monotonicity StanleyMonotonicity
• Lemma 3.4
• proof